Question

The correct variation of gravitational potential V with radius r measured form the centre of earth of radius R is given by

• A.
• B.
• C.
• D. None of the above

Answer 1

The correct option is D

None of the above

$V_Q$ = $\frac{GM}{R}$, dV = -$\frac{G.\frac{M}{\frac{4}{3}\pi R^3}.\frac{4}{3}\pi x^3}{x^2}$

=-$\frac{GM}{R^3}$.x.dx

$\Delta V=+\frac{GM}{R^3}{\int }_R^{r}xdx=-\frac{GM}{R^3}\frac{1}{2}(R^2-r^2)$

$\therefore$ $V_P$ = $V_Q$ + $\delta V$ = -$\frac{GM}{R}$-$\frac{GM}{2R}$+$\frac{GM}{2R^3}{r}^2$.

= -$\frac{3GM}{2R}$+$\frac{GM}{2R^3}{r}^2$

Is the equaton of a parabola (assuming earth to be of uniform density)